Finite Group Databasebeta

← #10604884

Subgroups — #10604884

Order 640.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D5C211yesmaximal, characteristic
2#65756C212yesmaximal, complemented
3#64933C212yesmaximal, complemented
4#64914C212yesmaximal, complemented
5C23×C4×C5C411yescharacteristic
6C22×C22⋊D5C411yescharacteristic
7C22×C4×D5C2211yescharacteristic
8C2×#700—22no—
9C2×C22⋊Dic5C2211yescharacteristic
10C4×C22⋊D5—22no—
11C4×C22⋊D5—22no—
12C2×#685C411yescharacteristic
13C2×#683C2211yescharacteristic
14C2×C4×Dic5C411yescharacteristic
15C22⋊#170C2214yes—
16#4003—22no—
17#1465—51nomaximal, Sylow
18C24×C5C2×C411yescharacteristic
19C23×D5C2×C411yescharacteristic
20C22×C4×C5—21no—
21C22×C4×C5—21no—
22C22×C4×C5—42no—
23C22×C4×C5C2×C411yescharacteristic
24C22×C4×C5C2×C411yescharacteristic
25C2×C22⋊D5—22no—
26C2×C22⋊D5—22no—
27C2×C22⋊D5—22no—
28C22×Dic5C2×C411yescharacteristic
29C22×Dic5C2×C411yescharacteristic
30C22×Dic5C2311yescharacteristic
31C2×C4×D5—42no—
32C2×C4×D5D412yes—
33C2×C4×D5D412yes—
34C2×#170—24no—
35#700—44no—
36C22⋊Dic5—22no—
37#685—22no—
38#683—22no—
39C4×Dic5—21no—
40C4×Dic5—21no—
41C2×C4×D4—51no—
42#407—52no—
43#324—52no—
44#323—52no—
45C23×C5—21no—
46C23×C5—21no—
47C23×C5—41no—
48C23×C5—41no—
49C23×C5M1612yescomplemented
50C23×C5C22×C411yescharacteristic
51C22×D5—21no—
52C22×D5—21no—
53C22×D5—41no—
54C22×D5—41no—
55C22×D5C22⋊C412yes—
56C2×C4×C5—22no—
57C2×C4×C5—21no—
58C2×C4×C5—42no—
59C2×C4×C5—42no—
60C2×C4×C5—41no—
61C2×C4×C5C22⋊C411yescharacteristic
62C2×C4×C5C22⋊C411yescharacteristic
63C22⋊D5—28no—
64C2×Dic5—21no—
65C2×Dic5—21no—
66C2×Dic5—21no—
67C2×Dic5—41no—
68C2×Dic5—41no—
69C2×Dic5M1612yescomplemented
70C2×Dic5C2×D412yes—
71C4×D5—24no—
72C4×D5—42no—
73#170—44no—
74C22×D4—51no—
75C23×C4—51no—
76C23×C4AGL(1,𝔽5)11yescharacteristic
77C2×M16—102no—
78C4×D4—102no—
79C4×D4—102no—
80C2×C4⋊C4—51no—
81C2×C22⋊C4—51no—
82C2×C22⋊C4—51no—
83C2×C42—51no—
84C22⋊C8—54no—
85C22⋊C8—102no—
86C22×C5—21no—
87C22×C5—21no—
88C22×C5—22no—
89C22×C5—42no—
90C22×C5—42no—
91C22×C5—41no—
92C22×C5—41no—
93C22×C5C2×C22⋊C411yescharacteristic
94C22×C5C2×M1611yescharacteristic
95C22×C5C2×M1611yescharacteristic
96C2×D5—22no—
97C2×D5—22no—
98C2×D5—22no—
99C2×D5—42no—
100C2×D5—42no—
101C4×C5—21no—
102C4×C5—41no—
103C4×C5#9912yes—
104Dic5—22no—
105Dic5—22no—
106Dic5—41no—
107C24—51no—
108C24C2×AGL(1,𝔽5)11yescharacteristic
109C2×D4—102no—
110C2×D4—102no—
111C2×D4—102no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—42no—
115C22×C4—51no—
116C22×C4—51no—
117C22×C4—52no—
118C22×C4—52no—
119C22×C4—51no—
120C22×C4—202no—
121C22×C4C2×AGL(1,𝔽5)11yescharacteristic
122C22×C4C2×AGL(1,𝔽5)11yescharacteristic
123M16—204no—
124C2×C8—104no—
125C4⋊C4—102no—
126C22⋊C4—102no—
127C22⋊C4—102no—
128C42—101no—
129C42—101no—
130C2×C5—21no—
131C2×C5—21no—
132C2×C5—42no—
133C2×C5#40711yescharacteristic
134C2×C5C2×#9911yescharacteristic
135C2×C5#41311yescharacteristic
136D5—42no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—52no—
142C23—101no—
143C23—101no—
144C23—201no—
145C23—201no—
146C23#70012yescomplemented
147C23C22×AGL(1,𝔽5)11yescharacteristic
148D4—108no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—52no—
155C2×C4—52no—
156C2×C4—101no—
157C2×C4—101no—
158C2×C4—104no—
159C2×C4—101no—
160C2×C4—202no—
161C2×C4—201no—
162C2×C4—201no—
163C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
164C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
165C8—204no—
166C5#146511yescharacteristic, complemented, Sylow
167C22—21no—
168C22—21no—
169C22—22no—
170C22—42no—
171C22—42no—
172C22—41no—
173C22—41no—
174C22—102no—
175C22—102no—
176C22—102no—
177C22—202no—
178C22—202no—
179C22C2×#70011yescharacteristic
180C22D5⋊M1611yescharacteristic
181C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
182C4—21no—
183C4—41no—
184C4—102no—
185C4—102no—
186C4—201no—
187C4#401112yes—
188C2—21no—
189C2—21no—
190C2—42no—
191C2—202no—
192C2#6575611yescharacteristic
193C2#6576111yescharacteristic
194C2C2×#401111yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C4
FrattiniC23C22×AGL(1,𝔽5)
FittingC23×C4×C5C4
socleC22×C5C2×C22⋊C4
prime radicalC22×C22⋊D5C4
cosocleC22×Dic5C23
hypercenterC23×C4AGL(1,𝔽5)
hypocenterC5#1465
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #65756C2semidirect
3. #64933C2semidirect
4. #64914C2semidirect
49. C23×C5M16semidirect
69. C2×Dic5M16semidirect
146. C23#700semidirect
166. C5#1465semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×C22⋊D52
#657562
#649332
#649142
#14655